# PersistentHomologyTransfer.jl

Persistent Homology Transform is produced and maintained by

Yossi Bokor and Katharine Turner

yossi.bokor@anu.edu.au and katharine.turner@anu.edu.au

This package provides an implementation of the Persistent Homology Transform, as defined in Persistent Homology Transform for Modeling Shapes and Surfaces. It also does Rank Functions of Persistence Diagrams, and implements Prinipal Component Analysis of Rank functions.

## Installation

Currently, the best way to install PersistentHomologyTransfer is to run the following in `Julia`

:

```
using Pkg
Pkg.add(PersistentHomologyTransfer)
```

## Functionality

- PersistentHomologyTransfer computes the Persistent Homology Transform of simple, closed curves in \(\mathbb{R}^2\).
- Rank functions of persistence diagrams.
- Principal Component Analysis of Rank Functions.

### Persistent Homology Transform

Given an \(m \times 2\) matrix of ordered points sampled from a simple, closed curve \(C \subset \mathbb{R}^2\) (in either a clockwise or anti-clockwise direction), calculate the Persistent Homology Transform for a set of directions. You can either specify the directions explicity as a \(n \times 2\) array (`directions::Array{Float64}(n,2)`

), or specify an integer (`directions::Int64`

) and then the directions used will be generated by

```
angles = [n*pi/(directions/2) for n in 1:directions]
directions = [[cos(x), sin(x)] for x in angles]
```

To perform the Persistent Homology Transfer for the directions, run

`PHT(points, directions)`

This outputs an array of Eirene Persistence Diagrams, one for each direction.

### Rank Functions

Given an Eirene Persistence Diagram \(D\), PersistentHomologyTransfer can calculate the Rank Function \(r_D\) either exactly, or given a grid of points, calculate a discretised version. Recall that \(D\) is an \(n \times 2\) array of points, and hence the function `Total_Rank_Exact`

accepts an \(n \times 2\) array of points, and returns a list of points critical points of the Rank function and the value at each of these points. Running

`rk = Total_Rank_Exact(barcode)`

we obtain the critical points via

`rk[1]`

which returns an array of points in \(\mathbb{R}^2\), and the values through

`rk[2]`

wich returns an array of integers.

To obtain a discrete approximation of a Rank Function over a persistence diagram \(D\), use `Total_Rank_Grid`

, which acceps as input an Eirene Persistence Diagram \(D\), an increasing `StepRange`

for \(x\)-coordinates `x_g`

, and a decreasing `StepRange`

for \(y\)-coordinates `y_g`

. The `StepRanges`

are obtained by running

```
x_g = lb:delta:ub
x_g = ub:-delta:lb
```

with `lb`

being the lower bound so that \((lb, lb)\) is the lower left corner of the grid, and `ub`

being the upper bound so that \((ub,ub)\) is the top right corner of the grid, and \(delta\) is the step size.

Finally, the rank is obtained by

`rk = Total_Rank_Grid(D, x_g, y_g)`

which returns an array or values.

### PCA of Rank Functions

Given a set of rank functions, we can perform principal component analysis on them. The easiest way to do this is to use the wrapper function `PCA`

which has inputs an array of rank functions evaluated at the same points (best to use `Total_Rank_Grid`

to obtain them), an dimension \(d\) and an array of weights `weights`

, where the weights correspond to the grid points used in `Total_Rank_Grid`

.

To perform Principal Component Analysis and obtain the scores run

`scores = PCA(ranks, d, weights)`

which returns the scores in \(d\)-dimensions.

## Examples

### Persistent Homology Transfer

We will go through an example using a random shape and 20 directions. You can download the CSV file from here

To begin, load the CSV file into an array in Julia

`Boundary = CSV.read("`")
Persistence_Diagrams = PHT(Boundary, 20)

You can then access the persistence diagram corresponding to the \(i^{th}\) direction as

`Persistence_Diagrams[i]`